Artificial Intelligence: A Modern Approach
3rd Edition
ISBN: 9780136042594
Author: Stuart Russell, Peter Norvig
Publisher: Prentice Hall
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Chapter 7, Problem 16E
Explanation of Solution
Smallest set of clauses
- A clause is a disjunction of literals, and its models are the union of the sets of models
of each literal; and each literal satisfies half the possible models...
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Students have asked these similar questions
Let P(x) and Q(x) be predicates and suppose D is the domain of x. For the statement forms in the given pair, determine whether they have the same truth value for every choice of P(x), Q(x), and D, or whether there is a choice of P(x), Q(x), and D for which they have opposite truth values.
∃x ∈ D, (P(x) ∧ Q(x)) and (∃x ∈ D, P(x)) ∧ (∃x ∈ D, Q(x))
Use a direct proof technique to prove the following theorems:
For all integers x and y, x2 + y2 − 3 is not divisible by 4.
The sets A, B, and C are arbitrary subsets of some universal set U. Prove
that (C − A) ∪ (C − B) = C − (A ∩ B).
What is the Truth value of the Expression ∀ (x)P(x) in each of the following interpretation?
P(x) is the property that X is yellow, and the domain of interpretation is the collection of all buttercups.
P(x) is the property that x is yellow, and the domain of interpretation is the collection of all flowers.
P(x) is the property that x is plant, and the domain of interpretation is the collection of all flowers.
P(x) is the property that x is either positive or negative , and the domain of interpretation
Chapter 7 Solutions
Artificial Intelligence: A Modern Approach
Ch. 7 - Suppose the agent has progressed to the point...Ch. 7 - (Adapted from Barwise and Etchemendy (1993).)...Ch. 7 - Prob. 3ECh. 7 - Which of the following are correct? a. False |=...Ch. 7 - Prob. 5ECh. 7 - Prob. 6ECh. 7 - Prob. 7ECh. 7 - We have defined four binary logical connectives....Ch. 7 - Prob. 9ECh. 7 - Prob. 10E
Ch. 7 - Prob. 11ECh. 7 - Prob. 12ECh. 7 - Prob. 13ECh. 7 - Prob. 14ECh. 7 - Prob. 15ECh. 7 - Prob. 16ECh. 7 - Prob. 17ECh. 7 - Prob. 18ECh. 7 - A sentence is in disjunctive normal form (DNF) if...Ch. 7 - Prob. 20ECh. 7 - Prob. 21ECh. 7 - Prob. 23ECh. 7 - Prob. 24ECh. 7 - Prob. 25ECh. 7 - Prob. 26ECh. 7 - Prob. 27E
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- Suppose propositional sentences and knowledge-base here. We have a knowledge- base called KB. KBPrime is defined as an union of (1) KB and (2) the negation of a propositional predicate P. Answer true/false to the following questions. Note that "|=" is used to denote entailment, and "|-" is used to denote derivation through resolution. (a) resolution) P. (b). (c).. (d). If KB |= P, then KB |- P. That is, if KB entails P, then KB derives (using If KB |- P, then KB |= P. If KBPrime |- [], then KBPrime is not satisfiable. 40 If KBPrime is not satisfiable, then KBPrime |- [].arrow_forwardLet p1,p2,p3 and p4 be arbitrary propositions. Prove that p1, p2, p3 and p4 are all equivalent if p1 → p4, p2 → p1, p4 → p2, p3 →p4,andp4 →p3.arrow_forwardConsider the following list of sentences about members of the set P of all pets. 1. If any pet is not a black cat, then that pet has a tail. 2. If some pet has a tail, then that pet is a black cat. 3. At least one pet is a black cat. 4. Every pet which is a black cat has a tail. 5. Every pet which has a tail is a black cat. 6. If a pet is a black cat, then that pet does not have a tall. 7. Every pet which is a black cat does not have a tail. 8. At least one pet is a black cat and has a tail. Let b (x) be the predicate x is a black cat and let t (x) be the predicate x has a tail. 1) Va € P, b (x) →t (2) 2) Væ € P, t (x) → b(æ) 3) 3a € P|b (x) A t (z) 4) Væ € P, -b (x) →t (x) 5) Va € P, b (r) → t(2) 6) Jr € P|b(2)arrow_forward
- Let p1, p2, p3 and p4 be arbitrary propositions. Prove that p1, p2, p3 and p4 are all equivalent if p1 → p4, p2 → p1, p4 → p2, p3 → p4, and p4 → p3.arrow_forwardThe sentence P is truth functionally true if and only if Select one: O i. (P} has a closed truth tree. O ii. (P} has open branches. O iii. None of these O iv. (-P} has a closed truth tree. O v. (P} and (-P} have open branchesarrow_forwardLet p, q and r be propositions: p: Jasper has sore eyes q: Jasper misses the last day of the board exam r: Jasper passes the board exam Write the following propositions in terms of the given connectives: a.) q→¬r b.) q ↔ p c.) p → q ⋀ ¬rarrow_forward
- in theory graph Define and give examples to all : what is independent sets ? what is maximal independent set ? what is the independence number? what is the lower independence number? what is maximum independent set ?arrow_forwardQUESTION 3 Let F(x.y): "x is friend with y", N(x): "x is funny", W(x): "x is 'wise", R(x). "x is fair". The domain for x and y is the set of all persons. Write the english of: 3x Vy[¬N(x) A W(x)]A [R(y)→¬F(x,y)] For the toolbar, press ALT+F10 (PC) or ALT+FN+F10 (Mac).arrow_forwardA student wants to prove by induction that a predicate P holds for certain nonnegative integers. They have proven that for all integers n ≥ 0 that P(n) → P(3n). Suppose the student has proven P(3). Which of the following propositions can they infer? (The domain for any quantifiers appearing in the answer choices is the natural numbers.) O Vn, P(3+3) O P(n) does not hold for ʼn < 0 P(n) for n = 6, 9, 12,... Vn ≥ 1, P(3¹)arrow_forward
- Consider the following language: X = {(A)| A is a propositional sentence that has at least 2 valuations that make it true} Is X decidable? Justify your answer.arrow_forwardLet P (x) and Q(x) be two predicates and suppose D is the the domain of For the statement forms in each pair, determine whether they have the same truth value for every choice of p(x), Q(x) and D, or not. ∀x ∈ D, (P (x) ∧ Q(x)), and (∀x ∈ D, P (x)) ∧ (∀x ∈ D, Q(x)) ∀x ∈ D, (P (x) ∨ Q(x)), and (∀x ∈ D, P (x)) ∨ (∀x ∈ D, Q(x))arrow_forwardProposition (Distributive Law): For expressions P1, P2, P3, any word matching the regular expression (P1 (P2|P3)) also matches the regular expression ((P1P2) (P1P3)) Give a proof of the above proposition, or demonstrate that it is false.arrow_forward
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